The instantaneous velocity is the specific rate of change of position (or displacement) with respect to time at a single point #(x,t)#, while average velocity is the average rate of change of position (or displacement) with respect to time over an interval.
Graphically, the instantaneous velocity at any given point on a function #x(t)# is equal to the slope of the tangent line to the function at that location. Meanwhile, the average velocity is equal to the slope of the secant line which intersects the function at the beginning and end of the interval.
Typically, when confronted with a problem, it will be fairly evident whether instantaneous velocity or average velocity is called for. For example, suppose Timothy is moving along a track of some kind. Assume that Timothy's displacement function #x(t)# can be modeled as #x(t) = t^2 - 5t + 4#. In interests of simplicity, specific units shall be omitted.
If asked to find Timothy's velocity at a given point, instantaneous velocity would fit best. Thus, at a given point #(t_0, x_0)#, we differentiate our function with respect to #t#. The power rule informs us that #x'(t) = 2t -5#. (Recall that the derivative, or rate of change, of position (or displacement) with respect to time is simply velocity). Thus, for our given point, #x'(t_0) = 2(t_0)-5#
If asked to find Timothy's average velocity over the course of #b# units of time (starting at #t=0#), the calculation is easier in that we do not need derivatives. Instead, our average velocity will be represented by the difference between our #x# values at the endpoints, divided by the elapsed units of #t#. In other words, our general average velocity formula on an interval from #t=a# to #t=b# is
#v_(avg) = [x(b)-x(a)]/(b-a)#
In this case, #v_(avg) = [x(b)-x(0)]/(b-0) = [x(b) - x_0]/b#
Note that this leads to dividing by zero in cases where #a=b#. As may be evident, in such an event, we will be finding instantaneous velocity instead of average velocity.
